PAGE 9

.. 2007 . .. ( )

.

, , , , , . , , , . , , , , .

.

.23 , - 7.

.

.. 41. . 4 1.1. ............ 4 1.2. . . X S2 .. 51.3., , ..7

1.4. .91.5. ..............111.6. ......121.7. ...141.8. 161.8.1. ...161.8.2. , ..181.9. . .241.9.1. 24

2. ..242.1. .242.2. ....262.3. . R2..27 2.4. ..292.5. 312.5.1. ..332.5.2. ...352.6.. .37 2.6.1. . .392.7..42 3. .. 433.1 .................433.1.1. ..443.1.2. .

463.2. ...463.2.1. ( ) ..473.3. . 48 3.4. ...503.4.1. ..503.4.2. 1- .513.4.3. 2 .513.5. 54

3.5.1. ..543.5.2. ...553.5.3. ...56 3.5.4. .573.6. 583.6.1. ...60 4. ........62 ..77 . , , , , - ? , , - , , . , . , , , . , , .

.

, . . ( ) , , , . .1. .

1.1. (). - , . , , , . (, , , ) , , . , 1/2 , 1/6. ( ) (P ) , . P (), , . . , , . , ( ). (). . , ..( ). , , . , , , . , , . , ( ), , , . , () .1.2. . . X S2 .

- , . ( 1/6 ). . x, M(x)., x n (x1 x2 x3 x4 .. xn) xi Pi ,

M(x)= x1 p1 + x2 p2 + .. + xn pn = ( 1.1 ) . x . x , . x , . . (x - ), x. 2 , , , Var(x).

2 = Var(x)= M{(x - )2} (1. 2 )

2 , 2 , . x ,

x = + u, ( 1.3 )

u ( ) ,

u = x - (1.4)

M(u )= 0. (1.5)

x u, , x u.

2 = 2u = M(u2 ) (1.6) , 2 x u. , ( ), . , . . . n , .

, , , , , , .

: 2 :.1.1.

, X = 1/ n( xi )

, 2S2 = 1/( n- 1) { ( xi x )}

X Xi ()

S2

X , x .

, , X :

X = 1/ n (x1 + x2 + x3 + .+ xn ).

, xi = + ui, - x = + u - u - - ui. , , - () u ( ) . (),

,( ) .1.

x , , xi ui .

S2 - , xi - x =( + ui ) ( + u) = ui u. , S2..

1.3., , .

, . , , , (u) .

, , 2 S2 . , . .

, . , () , S2 Q2. , . - .

, , , , () 1.2.

, , . .

, .

, , (), () 1..3. , , . ?.

, . , , . , , . , .

, , , .

n?. Q2/n, , , (1.4). , n Q/n . :

Lim x =

n

, . , (1..5. ).

1.4. .

, , .

. . n (,) :

Cov(x,y)= (xi x ) (yi y )= { (x1 x ) (y1 y ) +.........+ (xn x )(yn y) (1.7) Cov(x, y) cov (x, y ) y .

Var( x ) var (x,).

, .

1. y = v +w Cov(x,y ) = Cov(x,v) + Cov(x,w) (1.8)

2.

y = ar, , Cov(x,y ) = a Cov(x,r) (1.9)

3.

y = , Cov(x,y )= 0 (1.10)

, y .

cov(x,y ) = y = M {(x -x ) (y - y ) } (1.11)

- -x , y .

, , . ,

M {Cov(x,y) } = { cov(x,y ) } (1.12)

, n/(n-1)

, .

. , , . n 1 n, : Var( x ) =( xi x )2 = [( x1 x )2 + ( x2 x )2 + ..... + (xn x )2] (1.13)

:

1. n/(n-1) Cov (x,y), s2 = n/ (n-1)[Var(x )]

, .

1.

y = v +w, Var( y ) = Var( v ) + Var( w ) + 2Cov (v,w) (1.14)

2.

y = z, , Var( y ) = a Var( z ) (1.15)

3. y = , , , Var( y ) = 0 (1.16)

4.

y = v +, , , Var( y ) = Var( v ) (1.17)

, .

Var( x ) = Cov (x,x) (1.18)

, . .

.

: xy = cov(x,y)/ var (x ) var (y) = ,y / 2 2y (1.19)

=0 . , - , =1, = -1. : rxy :

(1.20) , r 1, , - -1. r=0. r=0 , =0 . rxy xy. xy rxy 0,3, . 1.5. .

, , , . , , .

:

(1.21)

y, , : 1. + x, ( ) , ; 2. u. .6 .

1 , 2 , 3 , 4 , - . , Q1 Q2 Q3 Q4 . ( u.) , - 1 ,2 , 3 , 4. , , 1.6. , Q1 Q2 Q3 Q4 , .

, , .

, u . u =0, Q . .1.6. .

: .

, 1.7 , (1.21). , . 1.7 .

, , , b.

, . , , , , .

b , . (1. 8.) = 1 R1 , *1, 1. *1 1 1. (1 *1) P1 R1 . 1 . 2 3 4 . , , . , .

S. 1.8. :

S= 21 + 22 + 23 + 24 (1.22)

S b, . S , . S =0, , , 0.

. .1. , : = 3, = 1, = 5 = 2. b

( !.23), , , , . .

= 1, . = 2, ., 1.2.xyy*e

13 +b3 b

25 +2b5 2b

, . 1 , , (3 a b), 2, , (5 a 2b), :

, . , :

, :

, =1 =2. :

.

, , :

, . . 2. , : . , . .

, 1,2,3, , (a+2b) (+3b); .3.xyy*e

13

3-a-b

25(a+2b)5-a-b

36(a+3b);6-a-b

,

:

, =1,67 b=1,50. , :

1.7.

, n . , ,

y = + bx (1.24)

(.. ), . S= 21 + 22 + 23 + 24

, , :

b = Cov (x, y, )/ Var (x ) (1.25) a = y bx (1.26)

.

y y , , x. , *i i *i = i + i (1.27)

*i i - , , . , , , . i . , . (1.14) . Var ( ) = Var (* + )= Var (* ) + Var ( e ) + 2 Cov (*, e ) (1.28)

, , Cov ( y,e ) , .

Var ( ) = Var (* ) + Var ( e ) (1.29)

, Var ( ) : Var (*) Var ( e ) - .

*: Var ( + bx);

Var [- ( + bx)]. .

Var (* ) /Var ( ) , . (R2).

R2= Var (*) /Var (), (1.30)

R2 = 1 - Var () /Var (), (1.31)

(TSS) .

(SS) - (+bx) ( + bx ). ( ) (RSS) .

, R .

R= Var ( + bx ) /Var ( y ) = SS / TSS (1.32)

: TSS = RSS + SS.

R 1. , , , *i= i , i .

Var (* ) =Var (), Var ( )=0 R2= 1.

1.8.

b, , : 1) , ; 2) , Cov ( x, u ), b .

u.

1.8.1

, , . , - ( ).

.

, . , . , , , , . , , . , , . .

? , ( ?). . , . . - :

1) ;

2) ;3) ( ) u .

-, y = + x +u , ,x, u.

- , b y x. , b , . . , , , . , , .. y x. , y., , . , , , , .

1. (u )=0 i.

2. : 2 (ui ) i.

3.

Cov (ui, uj )=0, i j .

4. Cov (uk, k )=0 k.

: , , .

. b :

(1.33)

(1.34)

. , 2u. , , . , , .

2u :

(1.35)

:

(s.d.) ; .

(..) , .

(1.33), (1.34), b .

(1.36)

(1.37)

, :

1. , .2. n, Var ( x ) 2u (, ).3. :

(1.38)4. :

(1.39)1.8.2. ,

, .

. , , , . . ( ) . 00,1 0,05 1 5. , , . . (Q* ), (Q* + ), () .

( ) Q Q* , Q - Q* , Q -= Q* . . , . , 0,95; 0,99 0,999.

, . , , (., ) , ( w., ), :

. = = + w. +u, (1.40)

, u .

, , . , , , , 0 , =1 = 0. , 1, , , 0. , 1. :

0 : =1.

1 : 1. , , .

, .

(0 ) , .

(1) , , .

:

= + +u (1.41)

, . , , 0; 0 : = 0 1 : 0. , . , 4 - .

0 , , , 0 2(u ) /n Var ( x ), , u . , b , 1.9 .

, s.d.(b).= 2(u) /n Var (x ) 0 ( ). , b 0,1. , 1.10. , 0,8 1,2.

. , , , . 1, . , , 1, 0,7, 1. , , 0,0027, .., . , .

1. , .

2. , . , , 1, .. - 1 : 0. , , - ? , =0,7, , , . ?

. 5% 1%. 5%, , b 5%. , 5- .

, b 0 1.96 (s.d.). , , , b 1,96 s.d., 2,5% , - 2,5%. , 1,96 s.d.

, , ,

Z < 1, 96 Z < -1, 96 (1.42)

Z ; Z = b 0 / s.d. ( b) (1.43) :

0 1,96 s.d. ( b) < b < 0 + 1,96 s.d. ( b) (1.44)

b, , = 0. b 5% .

, s.d. ( b) = 0,1, 5%, b 0,196, .. 1,196 0,804. b 0,804 1,196. . 1.11.

, 1%.

0 2,58 s.d. ( b) < b < 0 + 2,58 s.d. ( b) (1.45) , 5% 1% . , 1 2 . , . , .

, , 1 . , 5%, 5% . 1%, 1% .

, 1% .

, , , , , . , . , 2 , . , 1 , .

, .

, b ?

, b . . b , : . Z .. (b ) s.d. ( b) t- ( ):

t = (b 0))/ ..(b) (1.46) , t , t . , .

. .

. 2 , n -2.

t , 1,96 5% -2,58 1% . , , = 0 :

- t . < ( b 0 )/ ..(b) < t . (1.47) t , .

:

, , , ( ) p. = -1,21 + 0,82w., .. ( 0,05) (0,10)

, , , , w -1,0. t- :

t = (b 0 )/ ..(b) = 0,82 1,00/ 0,10 = - 1,8

, , 20 , 18, t 5% 2,101. t- 2,101 2,101, , . = 0,82, 1,00, , .

, , :

b 0 / ..(b) t . < < b + 0 / ..(b) t . (1.48) , , b, , . , , .

5% , 95- , 1% - -99% .

t , 5% ,, 99% 95%. b , 99% 95% , .

:

b 0,093. 0,003, t 5% 2.069. 95% :

0.093- 0,003 2,069 0.093+ 0,003 2,069

, 0,087 0,099

0,099 0,087. , , .

.

, , , .

0 : R2 = 0 ( .. ) F ( ). :

F F ,

( 0 : r = 0 ) r t- :

(1.49) , 5%, t (n-2) . t ( ), , r =0, , ( ).

0 : = 0 0 : R2 = 0 . 0 : r = 0. .

1.9. . = f (x), z = g(x), F (z ) . , .

y= +/x y= x . , . , ( ) ( ). , . z = 1/x, y= +z . , z .

, :y= x u (1.50)

y*=lny, z=lnx *=ln, (1.50), :

y*= * +z (1.50)

y= ax u

lny= lna +lnx +lnu (1.51)

1.9.1.

y= x . , , . :

= dy/dx = (1.52) , :

y=0,01X 0,30 , (1.53) , 0,3. , () 1% () 0,3%.

2. .

2.1. .

, .

, .

: y = +1x1 + 2x2 + 3x3......... + ........... m x m + u (2.1) y = 1x1 + 2x2 + 3x3......... + ........... m x m + u (2.2) . : 1 2 3............ m , .

, , . S, .. :

S = 21 + 22 + 23 + + 2n - i, *, , :

* = + b1 1 + b22 (2.3)i = i * = i - b1 1 - b2 2 (2.4) :

(2.4a) (2.5)

(2.6) , . , , , - TSS, - ESS, - RSS . .

- R2=ESS :TSS, - .

.

( ) ..(i ). :

(2.7)

(2.6)- , . .

2.2. .

. , . ,

(2.7) ,

y = +1z1 + 2z2 + .... (2.8) z1 = x12 , z2 =

. , , x , , , . ,

y= x p , (2.9) y , x - , p , - , , :

lgy= lg + lg x + lgp + lg (2.10) lgy, lg x lgp, lg x , lgp .

- . 1927 . , , , (Y), () (L), . , , :Y = AKaL1-a (2.11) , , , . log , , log L, (1 - ), .

L :Y/L=A (K/L)av (2.12)( v). , , :log (Y/L) = log A + a log (/ L) + log v (2.13) (Y), () (L) . 2.3. .

R2.

.

, 1 -1, , 1 -1.

, , () . , . , .. , , .

, , , . , . , . , .

, , : , ; . , , , . , . , .

, R2 , , = *. ( ) , . , 1 2 :

(2.14), , 1 , :

. (2.15) :

(2.16)

(2.14) (2.16) 1 2 , . (2.16) 2 0, , b2 , , . ( , .) , R2 (2.14), (2.16), . , R2 , , . , . , , , . ,- .

1 2 3............ m , . , .

F- , , , F- .

(2.17)- ESS- , RSS- () , k- , .

F , , . F , k , (n-k-1) (F-). F- , . F- , (2.17)) TSS ( ), , , . :

.2.1 , F- , ,

(...) (.)

(..)...,

..

F-

,

207,49369,1869,16/ 5,70=12,1

91,18165,7

F 3 16 1% 5,27; , F- , 12,1, .2.4. , , , .

, . , , .

.

1. , , , .

2. , , .

3. .

, .

.

, .

1. , , , . t .

2. , , . , - .

R2 , . , .

, , . , , , . , , . , , .

- , , . , , .

, . . ,

(2.18) , 1, (z) ,

(2.19.) , .

(2.20) 1. , , a . . , . , , .

, , :

(2.21) - ; - . , , . , () , . (), () () . :

(2.22) , (). , =0, . - (). , . , . ( ). . , ., (), (), , , , :

(2.23) t , , .

, , , , , , :

(2.24) , .2.5. , , , , .

, , , . , ?

, . - , : 1- 0 .

, , , ..

2.2 , 1974 . , . . , , , 2.1, - , - .

D, 1963-1973 .. 1974 -1982..

(2.25).2.2 1963- 1982..

( 1972., . .)196318,5196826,5197333,9197834,8

196419,7196926,7197425,5197932,9

196523,5197022,7197525,4198028,7

196623,6197128,0197631,1198129,6

196722,2197231,6197734,4198229,8

1963 1973.. D=0 :

(2.26) 1974- 1982.. D=1:

(2.27) , . y x . D, 11 , 9 , :

EMBED Equation.3 , 1963 1973., 2.1, 0,57, 1974- 1982..-3,83. t- , 5%. , :

5%.- , pgas, :

. . pgas 1% . R2 0,69 0 ,87. , () , .

R2 , t- . , ( ) .

2.5.1.

: . . :

- - . , 964 :

, , , 3400 7 . . : . :

(2.28) , , . , .. , .

:

(2.29) , , , . .2.2. . , -. , . - D.

(2.30)D=0 D=1 .

y,x :

103. 964 :

, , ?

: ( -0 ); 1; - 2; - 3. -0, . D1 , D2 , D3 :

0 D1 = D2 = D3 =0 1 D1 =1; D2 = D3 =0

0 D2 =1; D1 = D3 =0

0 D3 =1; D1 = D2 =0

:

, .

, . , .

- , . .

, , . :

(2.31)- .

- 1 0 (), - 2 0 , 3 0. 964 , :

.

, 1 127 0, 2- 102, 3 105. , , , .2.5.2.

, , , , 4 . . . , , . , - , , , .. 1 4- 6 . 1- . :

(2.32) D2, D3 D4 , : D2 , , D3=1 3 0 , D4=1 4 0 .

- , . - 2 1- , .

EMBED Equation.3 .

, :

. (..) (0,09) (0,005) (0,09) (0,10) (0,10)

:

2.5.3. .

, ( ), , , . , . () () (D):

(2.33) , , , .

, - D

(2.34)

(2.35) D=0, , .

D=1, , . () , , () , .

, , Dx . 964 :

D=0 D=1, :

( )

( , ).

, , , , , , (12,1 4,0), 5%.2.6..

. , . , . , . , . :

y = +x + u , :

1. (ui ) = 0, ;

2. 2 (ui ) = 0. ;

3. Cov (ui ,uj ) = 0, i = j;

4. Cov (i ,ui ) = 0, .

, , .

, .

, (. 2.3.).

, .

, . ,

:

: 2 (ui ) = 2 : 2 (ui ) = i2 , (.2.4.) - , .

.

. , , 2.4., ( 1 2.3), = + , , , . , ( n 2.3), . , , . , , , , , . . 2.6.1. . .

, . , . :

; ; .

, , . .

, , , , . , :

(2.36) Di -

, 0, 0 1/ ( n-1 ) . 5%, 1,96, 1%, 2,58. , .

, , ( ) (GDP)

, 1 . . 67 .. ( - ) :

. 2,58, , , . , , u , . n , ; ( n-2) . , u ,

, i i , , i= + xi (2.37) , , xi i. , i. , 0. ?

- u i. , , . :

(2.38)

(2.39) - , I

. . , . , , .

, z zi = - . z :

(2.40) , .

, , . , :

(2.41) , .

2.7.

- Cov (ui ,uj ) = 0, i = j; . .

.

, .

( 2.5)

- , (-2.6). , .

, :

Uk+1 = p uk + ek+1, p - , ek+1 -

, :

(2.42)

ek - .

, =0 DW . , DW , . - 1 -1, DW 0 4.

.

- , :

1. .

2. .

3. ek ek-1 Uk+1 = p uk + ek+1, ek-1, .4. yk = +xk + uk yk = gk +xk + uk , yk = yk - yk-1 , xk = xk - xk-1 gk =1 - ,

.5. , .

3. . 3.1 .

(...). :

X11(t) X12(t) X13(t) -------- Xnm (...) X21 (t) X1 2(t) X13(t) -------- Xnm(t) (3.1) ------------------------------------------ Xn1 (t) Xn 2(t) Xn3(t) -------- Xnm(t) Xnm - n - , n - t. , , t , . , ( .. ... ), . X(t1),(X t2), (X t3), .. (X tn) (t) t1 t2 t3 t4 tn. .. t2 t1 = t3 t2 =.. tn tn - 1 = , - . : (1) (2) (3).. (N). 3.1.1. . , , .

1 2 ..n , .

1. .

2. , ..

P { (X(t1) < X } P { (X(t2) < X }, t1 t2 (74) , , . , , :

. , ( ) X(t). f(t ), . .., . - (t). . . ( ), , , . (t). .

. () , . . ( , ) (t).

:

1. ( ) X(t).

2. , X(t) , , , . , , X(t) :

X(t) = X() f + X() (t) + X() (t) + (t), (3.2) t = 1,2.N , , , . , .

. , .

3.1. X(t) , 96 1963 1970 ., (.. t= 1,2.96; )

, . . 3.2. 1960 1971 ..(; , N = 48 ).

.3.3 56 ( 1884 1939..). , N = 56.

, ( , ).3.1.2. .

. X(t) , n X(t1 ),( X t2), (X t3), .( X tn ) n X(t1 + ),( X t2 + ), (X t3 + ), .( X tn + ), n , t1, t2...tn, . , t, , t , :

(t ) = a (3.3) D ( t ) = [ (t ) ]2 = 2 (3.4) X(t) , D ( t ) . 2 :

- (3.5) (3.6) ( ), - t.

, , .

.

3.2 .

3.2.1. , .

(t ) (t1 ) t. , (t ) : ( ) = cov[ (t ), X(t1 + ) ] = [ (t ) - , X(t1 + ) - ] (3.7) - ( ) . ( ) - :

(3.8) =1,2,. N

= , (0 )

(3.9) , . .

(3.10) r() r(). .

:

(3.11)3.2.2. ( ) .

, .

1- (t ), (t + 2), , , ( .. (t + 1) = )

(3.12) r02 = r([ (t ), X(t1 + 2 ) ]) (3.13) r01 = r([ (t ), X(t1 + 1 ) ]) (3.14) r21 = r([ (t+2 ), X(t1 + 1 ) ]) (3.15) r02= r(2), r01= r(1), r21= r(-1) = r(1), (3.16) . 2 - ( . . 3- , , 2- ) :

(3.17) .

3.3. .3.3.1. .

f , (t) (t)

X(t) = X() f + X() (t) + X() (t) + (t), (3.18) :

1. ( t) ; H0 : (t) = = const (3.19)

H1 : (t) = const (3.20)2. ( ) X(t) = X() f + X() (t) + X() (t), .. (t), X(t) ,. H0 , .

, ..

(1) (2) (3).. (n). (3.21) ( ) (1) (2) (3).. (n). n , n , X(t) +, X(t) > nmed - , X(t) > nmed , nmed . , . , (n) (n) . - . , (1) (2) (3).. (n) , ( .. ), + - . .. + -, , , . , . (n), (n). H0 : (t) = = const ;

(3.22) , , 0,05 0,0975 ( , , ). ( ) , , .

, + -, .

, (1) (2) (3).. (n)

i +, (i +1) - i > 0, - (i +1) - i < 0 ( , ). , ( ), ( ). : , , , .

(n), (n) - , , + .

:

(3.23)

, H0 .

.

, .

( ) , X(t) =X() f + X() (t) +X() (t),

t = 1,2,..n.. X(t) = f (t) + (t), f (t) -, (t) .

, ( 3.2 ), , t, ..

(3.24) - ( . , . ( ) , .

f (t) (t) ( .. ) .

. X(t) Q2 , 1 + 2 +3 +.. (N)/ N , Q2 / N . (t) : (3.25) - , , .. t m + 1 n-m , ( ).3.4.

3.4.1.

, (t) .. , .

X(t) =X() f + X() (t) +X() (t), t = 1,2,..n (3.26).. X(t) = f (t) + (t), f (t) , (t) -

, .

() (t), , ( ) .. 0., .. (t) 0. (t) :

(t) 0, (3.27)[ (t), (t )] = Q0 2 =0, 0, (3.28)

, Q0 2 , t.

, , : m (t)= (t) + 1 (t-1) + 2 (t-2) + .. = 1, (t - k)], (3.29) k =0 m

0 = 1, 2 k <

k =0

, , . , (t) , ( ), ..

(t) = 1 (t-1) + 2 2 (t-2) + . (t). = k (t - k) + (t). (3.30) k =1 1, 2, 3 , (t). .

3.4.2. 1- 1- , j, , 0 :(t) = 1 (t-1) + (t), (3.31) - , < 1 , (t) . ( .. , (3.27 3.28). . :

. (3.32) = .

. 3.4.3. 2 2 ( ), , j , , 0. :

(t) = 1 (t-1) + 2 2 (t-2) + (t), (3.33) (1), (2) .

1 2 :1 = r(1) [ 1- r(2) ] : [1- r2 (1)] (3.34)2 = r(2) r2 (1) ] : [1- r2 (1)] (3.35) (3.34,3.35) r(1) r(2):

(3.36)

(103).

.

1- 2

1

< 1 (3.37) 2 , 1 , 2 : 1 j =1,2,.......p.

- 1 2 3 .1 2 3 Q0 2 ( 1 )

(3.57) , ( 1 ) ( .. < 1 ). ( 2 ) ( 1, 2 ) .

= - 1 ( 1- 2 ) : ( 1 + 21 + 2 1 ) (3.58) = - 2 : ( 1 + 21 + 2 1 ) (3.59) 3. 5.

3 5. 1.

, . X(t) , , , , , . , (t) , X(t) f(t), ( ) . X(t). ( ) (p, ,k ) . X(t), t = 1,2........ N, :

) f, ( - t ) k-1(k< 1) .) X(t), t = 1,2........, N-k , X(t) - , (p, ). 3.5.2. . X(1), X(2), X(3), ............ X(N), (3.60)

y(1), y (2), y (3), ............ y (N), , y(t) X(t), X(t - 1), X(t 2) , t +1, < N. , :

y(t)= 0 + k (t - k) + (t). (3.61) k =0 (t) , t = 1,2,.. N Q0 2 = D (t) - ( X(t), X(t-1),....... X(t- T), ; 0 , 0, 1, T .

. , ( ) t k (k > 0), , k (). k k , . .

, , . . , (.3.4, ) (.20). .3.4.3.5 , , , .

. , (t - ) (t) T , , . ( T = ) . B= k , .. Lim k = B